Grade 3: Unit 3.G.A.1-2, Reason with shapes and their attributes

Overview: The overview statement is intended to provide a summary of major themes in this unit.

In this unit, students analyze, compare, sort, and classify two-dimensional shapes through various problem-solving experiences. They learn to apply their ideas to entire classes of shapes rather than to individual shapes. They reason about, compose, and decompose polygons to make other polygons. Students understanding of the properties of shapes continues to be redefined, as they understand that shared attributes can define a larger category. They investigate quadrilaterals and recognize shapes that are not quadrilaterals. Students in Grade 3 also sort geometric figures and identify squares, rectangles, and rhombuses as quadrilaterals. Students also partition shapes into parts with equal areas and express the area of each part as a unit fraction of the whole. Classroom discussions are a vital part of this unit, and students should do the majority of the talking. The teachers role is to ask questions, to help students redefine their thinking, and to allow students to develop their understanding through whole-class, small group, and independent activities. Students should be encouraged to use proper mathematical vocabulary.

Teacher Notes: The information in this component provides additional insights which will help the educator in the planning process for the unit.

Review the Progressions for Geometry at: http://commoncoretools.files.wordpress.com/2012/06/ccss_progression_g_k6_2012_06_27.pdf to see the development of the understanding of Kindergarten through Grade 6 Geometry as stated by the Common Core Standards Writing Team, which is also the guiding information for the PARCC Assessment development.

It is important to note that the Progressions for Geometry state: In this progression, the term property is reserved for those attributes that

indicate a relationship between components of shapes. Thus, having parallel sides or having all sides of equal lengths are properties. Attributes and features are used interchangeably to indicate any characteristic of a shape, including properties, and other defining characteristics (e.g., straight sides) and non-defining characteristics (e.g., right-side up).

When implementing this unit, be sure to incorporate the Enduring Understandings and Essential Questions as the foundation for your instruction, as appropriate.

Students should engage in well-chosen, purposeful, problem-based tasks. A good mathematics problem can be defined as any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific correct solution method (Hiebert et al., 1997). A good mathematics problem will have multiple entry points and require students to make sense of the mathematics. It should also foster the development of efficient computations strategies as well as require justifications or explanations for answers and methods.

Learning about Geometry does not progress in the same way as learning about number, where the size of the number gradually increases and new kinds of numbers are considered later. Instead, students reasoning about Geometry develops through five sequential levels in relation to understanding spatial ideas. In order to progress through the levels, instruction must be sequential and intentional. These levels were hypothesized by Pierre van Hiele and Dina van Hiele-Geldof. For more information about the van Hiele Levels of Geometric Thought listed below, please go to: http://images.rbs.org/cognitive/van_hiele.shtml.

Level 0: Visualization

Level 1: Analysis

Level 2: Informal Deduction

Level 3: Deduction

Level 4: Rigor

Attributes refer to any characteristic of a shape.

Through your discussions and interactions with students, emphasize reasoning about shapes and their attributes as emphasized in the Maryland Common Core Standards, as opposed to simply identifying figures, which is typically only a vocabulary exercise. Definitions of geometric terms should connect to and evolve from classroom experiences and discussions.

When partitioning shapes into parts with equal areas, express the area of each part as a unit fraction of the whole. Students are building on their understanding of fractional parts of the whole (the parts that result when the whole or unit has been partitioned into equal sized portions or fair shares) from first and second grade.

Students should be encouraged to develop ideas and definitions about properties and classes of shapes based on their own concept of developments. Only after they have had ample time to build and discuss their own ideas should formal definitions be introduced.

In the U.S., the term trapezoid may have two different meanings. Research identifies these as inclusive and exclusive definitions. The inclusive definition states: A trapezoid is a quadrilateral with at least one pair of parallel sides. The exclusive definition states: A trapezoid is a quadrilateral with exactly one pair of parallel sides. With this definition, a parallelogram is not a trapezoid. (Progressions for the CCSSM: Geometry, The Common Core Standards Writing Team, June 2012.)

Enduring Understandings: Enduring understandings go beyond discrete facts or skills. They focus on larger concepts, principles, or processes. They are transferable and apply to new situations within or beyond the subject.

Geometry helps us understand the structure of space and the spatial relations around us.

Through geometry we can analyze the characteristics and properties of two- and three-dimensional shapes, as well as develop mathematical arguments concerning geometric relationships.

Geometry helps us develop and use rules for two- and three-dimensional shapes.

Analyzing geometric relationships develops reasoning and justification.

Essential Questions: A question is essential when it stimulates multi-layered inquiry, provokes deep thought and lively discussion, requires students to consider alternatives and justify their reasoning, encourages re-thinking of big ideas, makes meaningful connections with prior learning, and provides students with opportunities to apply problem-solving skills to authentic situations.

Where in the real world can I find shapes?

How can objects be represented and compared using geometric attributes?

How can I put shapes together and take them apart to form other shapes?

How are geometric shapes and objects classified?

Content Emphasis by Cluster in Grade 3: According to the Partnership for the Assessment of Readiness for College and Careers (PARCC), some clusters require greater emphasis than others. The chart below shows PARCCs relative emphasis for each cluster. Prioritization does not imply neglect or exclusion of material. Clear priorities are intended to ensure that the relative importance of content is properly attended to. Note that the prioritization is in terms of cluster headings.

Key:

Major Clusters

Supporting Clusters

Additional Clusters

Operations and Algebraic Thinking

Represent and solve problems involving multiplication and division.

Understand the properties of multiplication and the relationship between multiplication and division.

Multiply and divide within 100.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Number and operations in Base Ten

Use place value understanding and properties of operations to perform multi-digit arithmetic.

Number and Operations Fractions

Develop understanding of fractions as numbers.

Measurement and Data

Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.

Represent and interpret data.

Geometric measurement: understand concepts of area and relate area to multiplication and addition.

Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.

Geometry

Reason with shapes and their attributes.

Focus Standards: (Listed as Examples of Opportunities for In-Depth Focus in the PARCC Content Framework document):

According to the Partnership for the Assessment of Readiness for College and Careers (PARCC), this component highlights some individual standards that play an important role in the content of this unit. Educators should give the indicated mathematics an especially in-depth treatment, as measured for example by the number of days; the quality of classroom activities for exploration and reasoning; the amount of student practice; and the rigor of expectations for depth of understanding or mastery of skills.

3.OA.A.3 Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

3.NF.A.2 Developing an understanding of fractions as numbers is essential for future work with the number system. It is critical that students at this grade are able to place fractions on a number line diagram and understand them as a related component of their ever-expanding number system.

3.MD.C.7 Area is a major concept within measurement, and area models must function as a support for multiplicative reasoning in grade 3 and beyond.

Possible Student Outcomes: The following list provides outcomes that describe the knowledge and skills that students should understand and be able to do when the unit is completed. The outcomes are often components of more broadly-worded standards and sometimes address knowledge and skills necessarily related to the standards. The lists of outcomes are not exhaustive, and the outcomes should not supplant the standards themselves. Rather, they are designed to help teachers delve deeply into the standards and augment as necessary, providing added focus and clarity for lesson planning purposes. This list is not intended to imply any particular scope or sequence.

The student will:

Understand concepts of area and relate area to multiplication and to addition.

Recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.

Understand that shapes in different categories may share attributes and that the shared attributes can define a larger category.

Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.

Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole.

Engage in well-chosen, purposeful, problem-based tasks that promote reasoning with shapes and their attributes.

Collaborate with peers in an environment that encourages student interaction and conversation that will lead to mathematical discourse about geometry.

Progressions from Common Core State Standards in Mathematics: For an in-depth discussion of the overarching, big picture perspective on student learning of content related to this unit, see:

The Common Core Standards Writing Team (23 June 2012). Progressions for the Common Core State Standards in Mathematics (draft), accessed at: http://commoncoretools.files.wordpress.com/2012/06/ccss_progression_g_k6_2012_06_27.pdf

Vertical Alignment: Vertical curriculum alignment provides two pieces of information: (1) a description of prior learning that should support the learning of the concepts in this unit, and (2) a description of how the concepts studied in this unit will support the learning of additional mathematics.

Key Advances from Previous Grades:

Students in Prekindergarten:

Match like (congruent and similar) shapes.

Group shapes by attributes.

Correctly name shapes (regardless of their orientations or overall size).

Students in Kindergarten:

Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in front of, behind, and next to.

Correctly name shapes regardless of their orientations or overall size.

Identify shapes as two-dimensional (lying in a plane, flat) or three-dimensional (solid).

Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts and other attributes.

Model shapes in the world by building shapes from components and drawing shapes.

Compose simple shapes to form larger shapes.

In Grade 1, students:

Distinguish between defining attributes and non-defining attributes.

Build and draw shapes to possess defining attributes.

Compose two-dimensional shapes or three-dimensional shapes to create a composite shape.

Compose new shapes from composite shapes.

Partition circles and rectangles into two and four equal shares.

Describe partitioned shares using the words halves, fourths, and quarters.

Use the phrases half of, fourth of, and quarter of.

Describe the whole as two of, or four of the shares.

In Grade 2, students:

Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces.

Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

Partition circles and rectangles into two, three, or four equal shares, and describe the shapes using the words halves, thirds, half of, a third of, etc.

Describe the whole as two halves, three thirds, four fourths.

Recognize that equal shares of identical wholes need not have the same shape.

Additional Mathematics:

In Grades 4 and beyond, students:

Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

Graph points on the coordinate plane to solve real-world and mathematical problems.

Classify two-dimensional figures into categories based on their properties.

Solve real-world and mathematical problems involving area, surface area, and volume.

Draw, construct, and describe geometrical figures and describe the relationships between them.

Solve real-life and mathematical problems involving angle measure, are, surface area, and volume.

Understand congruence and similarity using physical models, transparencies, or geometry software.

Possible Organization of Unit Standards: This table identifies additional grade-level standards within a given cluster that support the over-arching unit standards from within the same cluster. The table also provides instructional connections to grade-level standards from outside the cluster.

Over-Arching

Standards

Supporting Standards

within the Cluster

Instructional Connections outside the Cluster

3.GA.A.1 Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.

3.MD.D.8 Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.

3.GA.A.2 Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape.

3.NF.A.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

3.NF.A.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.

Connections to the Standards for Mathematical Practice: This section provides examples of learning experiences for this unit that support the development of the proficiencies described in the Standards for Mathematical Practice. These proficiencies correspond to those developed through the Literacy Standards. The statements provided offer a few examples of connections between the Standards for Mathematical Practice and the Content Standards of this unit. The list is not exhaustive and will hopefully prompt further reflection and discussion.

In this unit, educators should consider implementing learning experiences which provide opportunities for students to:

1. Make sense of problems and persevere in solving them.

a. Determine whether concrete or virtual models, pictures, mental mathematics, or equations are the best tools for solving the problem.

b. Check the solution with the problem to verify that it does answer the question asked.

c. Listen to the strategies of others and attempt various approaches.

2. Reason abstractly and quantitatively

a. Connect number quantities to written symbols.

b. Create logical representations of problems.

3. Construct Viable Arguments and critique the reasoning of others.

a. Explain thought processes used to solve problems and respond to others explanations and ideas.

b. Compare the models used by others with yours.

c. Examine the steps taken that produce an incorrect response and provide a viable argument as to why the process produced an incorrect response.

d. Construct arguments using concrete referents, such as objects, pictures, and drawings.

4. Model with Mathematics

a. Construct visual models using concrete or virtual manipulatives, pictures, or equations to justify thinking and display solutions.

b. Evaluate results in the context of the situation and reflect on whether the results make sense.

5. Use appropriate tools strategically

a. Use tools such as graph paper to find all the possible rectangles that have a given perimeter. Compile the possibilities into an organized list or a table, and determine whether they have all the possible rectangles.

b. Use pattern blocks, Geoboards, number lines, attribute blocks, geometric shapes, or other models, as appropriate.

6. Attend to precision

a. Use mathematics vocabulary such as rhombus, quadrilateral, rectangle, etc. properly when discussing problems.

b. Specify units of measure and state the meaning of the symbols chosen (e.g., record responses in square units when determining the area of a rectangle).

7. Look for and make use of structure.

a. Use structure of various types of quadrilaterals to explain their relationship.

8. Look for and express regularity in reasoning

a. Notice repetition and patterns illustrated mathematics to find various methods for problem solving.

Content Standards with Essential Skills and Knowledge Statements and Clarifications: The Content Standards and Essential Skills and Knowledge statements shown in this section come directly from the Maryland State Common Core Curriculum Frameworks. Clarifications were added as needed. Educators should be cautioned against perceiving this as a checklist. All information added is intended to help the reader gain a better understanding of the standards.

Standard

Essential Skills and Knowledge

Clarification

3.GA.A.1 Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.

Essential Skills and Knowledge

Ability to compare and sort polygons based on their attributes, extending beyond the number of sides (2.G.1)

Ability to explain why two polygons are alike or why they are different based on their attributes

Through classroom activities and discussions, students will come to understand that a quadrilateral is a four sided figure, and that parallelograms are quadrilaterals with opposite sides parallel and equal.

They should also understand that a rhombus is a parallelogram with four equal sides, and that a square is a rhombus with equal angles. They should continue discussions from earlier grades about whether or not a square is also a rectangle, and should begin making connections about shapes belonging to more than one class and to larger classes now.

Activities that promote students learning include but are not limited to:

Describing how shapes are similar and how they are different

Creating their own definitions of particular classes of shapes

Shape sorting and classifying

Materials that help promote students learning include but are not limited to:

Pattern blocks

Geoboards

Paper and pencil

Attribute blocks

Color tiles

Virtual manipulatives and other technology

3.GA.A.2 Partition shapes into. parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape.

Essential Skills and Knowledge

Knowledge that this is a geometry application of unit fractions (3.NF.1) and ability to make use of unit fraction understanding.

Ability to use concrete materials to divide shapes into equal areas (e.g., pattern blocks, color tiles, geoboards)

Students should be presented with activities that require them to apply previous understanding about partitioning shapes. The focus in grade three is on expressing the area of each equal part as a unit fraction. Students begin to understand that equal parts has a more precise meaning: parts with equal measurements. Activities should include area, length, and set models, and incorporate but not be limited to the following:

If the yellow hexagon = 1, then how many green triangles can you fit on the hexagon? (6)

What is the unit fraction the one triangle represents? ( )

Materials that help promote students learning include but are not limited to:

Pattern blocks

Geoboards

Paper and pencil

Attribute blocks

Color tiles

Virtual manipulatives and other technology

Evidence of Student Learning: The Partnership for the Assessment of Readiness for College and Careers (PARCC) has awarded the Dana Center a grant to develop the information for this component. This information will be provided at a later date. The Dana Center, located at the University of Texas in Austin, encourages high academic standards in mathematics by working in partnership with local, state, and national education entities. Educators at the Center collaborate with their partners to help school systems nurture students' intellectual passions. The Center advocates for every student leaving school prepared for success in postsecondary education and in the contemporary workplace.

Fluency Expectations and Examples of Culminating Standards: This section highlights individual standards that set expectations for fluency, or that otherwise represent culminating masteries. These standards highlight the need to provide sufficient supports and opportunities for practice to help students meet these expectations. Fluency is not meant to come at the expense of understanding, but is an outcome of a progression of learning and sufficient thoughtful practice. It is important to provide the conceptual building blocks that develop understanding in tandem with skill along the way to fluency; the roots of this conceptual understanding often extend one or more grades earlier in the standards than the grade when fluency is finally expected.

Students fluently multiply and divide within 100.

Students fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.

Common Misconceptions: This list includes general misunderstandings and issues that frequently hinder student mastery of concepts regarding the content of this unit.

Not realizing that when partitioning shapes in order to express the area of each part as a unit fraction, all parts must be equal.

Not realizing that some the properties of some shapes may place them into more than one larger category. For example, a square is a rectangle, parallelogram, and a quadrilateral.

The inability to create both regular and irregular polygons.

Not enough experience with the properties of shapes to recognize and correctly draw shapes having specified attributes, such as a given number of angles.

Interdisciplinary Connections: Interdisciplinary connections fall into a number of related categories:

Literacy standards within the Maryland Common Core State Curriculum

Science, Technology, Engineering, and Mathematics standards

Instructional connections to mathematics that will be established by local school systems, and will reflect their specific grade-level coursework in other content areas, such as English language arts, reading, science, social studies, world languages, physical education, and fine arts, among others.

Available Model Lesson Plan(s)

The lesson plan(s) have been written with specific standards in mind. Each model lesson plan is only a MODEL one way the lesson could be developed. We have NOT included any references to the timing associated with delivering this model. Each teacher will need to make decisions related to the timing of the lesson plan based on the learning needs of students in the class. The model lesson plans are designed to generate evidence of student understanding.

This chart indicates one or more lesson plans which have been developed for this unit. Lesson plans are being written and posted on the Curriculum Management System as they are completed. Please check back periodically for additional postings.

Standards Addressed

Title

Description/Suggested Use

3.G.A.1

3.G.A.2

Reason about Shapes & Their Attributes

Students will explore and discuss the properties of geometric shapes, with the focus on quadrilaterals. In addition, they will partition shapes into equal parts and express each part as a unit faction of the whole.

Grade 3: Unit 3.G.A.1-2, Reason with shapes and their attributes

July 25, 2013 Page 23 of 23

Available Lesson Seeds

The lesson seed(s) have been written with specific standards in mind. These suggested activity/activities are not intended to be prescriptive, exhaustive, or sequential; they simply demonstrate how specific content can be used to help students learn the skills described in the standards. Seeds are designed to give teachers ideas for developing their own activities in order to generate evidence of student understanding.

This chart indicates one or more lesson seeds which have been developed for this unit. Lesson seeds are being written and posted on the Curriculum Management System as they are completed. Please check back periodically for additional postings.

Standards Addressed

Title

Description/Suggested Use

3.G.A.1

Shape Scavenger Hunt

Students go on a Scavenger Hunt around their school recording all the shapes they see in the Math Journal. When they return to the classroom, they choose one shape to write about, explaining why it is that shape.

Sample Assessment Items: The items included in this component will be aligned to the standards in the unit and will include:

Items purchased from vendors

PARCC prototype items

PARCC public released items

Maryland Public release items

Formative Assessment

Topic

Standards Addressed

Link

Notes

Geometric Pictures of One Half

3.G.A.2

http://www.illustrativemathematics.org/illustrations/1061

This task presents students with some creative geometric ways to represent the fraction one half. The goal is both to appeal to students' visual intuition while also providing a hands on activity to decide whether or not two areas are equal. In order for students to be successful with this task, they need to understand that area is additive in the sense described in 3.MD.7d. Students should be given paper models of each picture which they can fold or cut and rearrange so as to help visualize why the shaded and unshaded areas are equal.

Representing Half of a Circle

3.G.A.2

http://www.illustrativemathematics.org/illustrations/1014

This task continues "3.G Which pictures represent half of a circle?" moving into more complex shapes where geometric arguments about cutting or work using simple equivalences of fractions is required to analyze the picture. In order for students to be successful with this task, they need to understand that area is additive in the sense described in 3.G.7.d. This task is meant for instructional purposes, and students should have access to multiple copies of the figures and be encouraged to experiment by cutting them and comparing the pieces. For the three pictures which do represent one half, this can be seen with a suitable rotation of the picture, which switches the shaded and unshaded portion, or by cutting out the shaded pieces and pairing them up with the unshaded ones so as to match in both shape and size. This task would not be appropriate for high-stakes assessment.

Halves, Thirds, & Sixths

3.G.A.2

3.NF.A.1

3.NF.A.3b

3.MD.C.6

http://www.illustrativemathematics.org/illustrations/1502

The purpose of this task is for students to use their understanding of area as the number of square units that covers a region (3.MD.6), to recognize different ways of representing fractions with area (3.G.2), and to understand why fractions are equivalent in special cases (3.NF.3.b). Determining the fraction of the area that is shaded for rectangles A-D in part (b) is increasingly complex. Rectangles E, F, and G show that there are many ways for

of the area to be shaded blue, which implies that there are many ways to represent the fraction with area.

Rectangle H requires students to see the equivalence of two fractions, neither of which is a unit fraction. Students get a chance to demonstrate what they have learned in part (b) by generating their own representations of fractions in parts (c) and (d).

Interventions/Enrichments: (Standard-specific modules that focus on student interventions/enrichments and on professional development for teachers will be included later, as available from the vendor(s) producing the modules.)

Vocabulary/Terminology/Concepts: This section of the Unit Plan is divided into two parts. Part I contains vocabulary and terminology from standards that comprise the cluster which is the focus of this unit plan. Part II contains vocabulary and terminology from standards outside of the focus cluster. These outside standards provide important instructional connections to the focus cluster.

Part I Focus Cluster:

partitioning: dividing the whole into equal parts.

share: a unit or equal part of a whole.

partitioned: the whole divided into equal parts.

measurement quantities: examples could include inches, feet, pints, quarts, centimeters, meters, liters, square units, etc.

decomposing: breaking a number into two or more parts to make it easier with which to work.

Example: When combining a set of 5 and a set of 8, a student might decompose 8 into a set of 3 and a set of 5, making it easier to see that the two sets of 5 make 10 and then there are 3 more for a total of 13.

Decompose the number 4; 4 = 1+3; 4 = 3+1; 4 = 2+2

Decompose the number =

composing: Composing (opposite of decomposing) is the process of joining numbers into a whole numberto combine smaller parts.

Examples: 1 + 4 = 5; 2 + 3 = 5. These are two different ways to compose 5.

whole: In fractions, the whole refers to the entire region, set, or line segment which is divided into equal parts or segments.

numerator: the number above the line in a fraction; names the number of parts of the whole being referenced.

Example: I ate 3 pieces of a pie that had 5 pieces in all. So 3 out of 5 parts of a whole is written:

The 3 is the numerator, the part I ate. The 5 is the denominator, or the total number of pieces in the pie.

denominator: the number below the line in a fraction; states the total number of parts in the whole.

Example: I ate 3 pieces of a pie that had 5 pieces in all. So 3 out of 5 parts of a whole is written:

The 3 is the numerator, the part I ate. The 5 is the denominator, or the total number of pieces in the pie.

fraction of a region: is a number which names a part of a whole area.

Example:

Shaded area represents or of the region.

fraction of a set: is a number that names a part of a set.

Example:

The fraction that names the striped circles in the set is

unit fraction: a fraction with a numerator of one. Examples:

benchmark fraction: fractions that are commonly used for estimation or for comparing other fractions. Example: Is 2/3 greater or less than 1/2?

improper fraction: a fraction in which the numerator is greater than or equal to the denominator.

mixed number: a number that has a whole number and a fraction.

(The area of this rectangle equals 12 square units.The area of this shape equals 7 square units)area: the number of square units needed to cover a region. Examples:

(5 sq. units)tiling: highlighting the square units on each side of a rectangle to show its relationship to multiplication and that by multiplying the side lengths, the area can be determined. Example:

(3 x 5 = 15The area is 15 sq. units.)

(3 sq. units.)

rectilinear figures: a polygon which has only 90 and possibly 270 angles and an even number of sides.

Examples of Rectilinear Figures:

Part II Instructional Connections outside the Focus Cluster

(2 rows of 4 equal 8 or 2 x 4 = 8 3 rows of 4 or 3 x 4 = 12)arrays: the arrangement of counters, blocks, or graph paper square in rows and columns to represent a multiplication or division equation. Examples:

equivalent fractions: different fractions that name the same part of a region, part of a set, or part of a line segment.

=

Resources:

Free Resources:

http://wps.ablongman.com/ab_vandewalle_math_6/0,12312,3547876-,00.html Reproducible blackline masters

http://lrt.ednet.ns.ca/PD/BLM_Ess11/table_of_contents.htm mathematics blackline masters

http://yourtherapysource.com/freestuff.html Simple activities to encourage physical activity in the classroom

http://www.mathsolutions.com/index.cfm?page=wp9&crid=56 Free lesson plan ideas for different grade levels

http://sci.tamucc.edu/~eyoung/literature.html links to mathematics-related childrens literature

http://www.nctm.org/ National Council of Teachers of Mathematics

www.k-5mathteachingresources.com Extensive collection of free resources, math games, and hands-on math activities aligned with the Common Core State Standards for Mathematics

http://elementarymath.cmswiki.wikispaces.net/Standards+for+Mathematical+Practice Common Core Mathematical Practices in Spanish

http://mathwire.com/ Mathematics games, activities, and resources for different grade levels

http://www.pbs.org/teachers/math/ interactive online and offline lesson plans to engage students. Database is searchable by grade level and content

http://www.k8accesscenter.org/training_resources/MathWebResources.asp valuable resource including a large annotated list of free web-based math tools and activities.

http://www.cast.org/udl/index.html Universal Design for Learning

http://engageny.org/wp-content/uploads/2012/05/Shifts-for-Students-and-Parents.pdf Information for parents and students about the Shifts associated with the CCSS.

http://havefunteaching.com/ Various resources, including tools such as sets of Common Core Standards posters.

http://michellef.essdack.org/links Numerous mathematics links.

http://illustrativemathematics.org/ Tasks that align with the MD CCSS.

http://www.aimsedu.org/Puzzle/categories/topological.html Puzzles to challenge students of various ages.

http://www.insidemathematics.org/index.php/home Mathematics resources for teachers.

http://www.graniteschools.org/depart/teachinglearning/curriculuminstruction/math/Pages/MathematicsVocabulary.aspx Common Core vocabulary and vocabulary cards.

http://www.nctm.org/standards/mathcommoncore/ Math Common Core Coalition

Math Related Literature:

Blaisdell, Molly. If You Were a Quadrilateral.

Notes: The author gets students thinking with statements such as, If you were a quadrilateral, you would have four straight sides. You could be a checkerboard, a kite, or a yoga math. What else could you be if you were a quadrilateral. The book offers opportunities for rich classroom discussions about quadrilaterals.

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Emberly, Ed. Picture Pie.

Notes: The author uses 4 simple shapes to show students how to draw various patterns, animals, and people.

Taylor- Cox, Jennifer. Sigmund Square Finds His Family.

Notes: Even though he has order and friends in his life, things are not fair-square for Sigmund Square. With its many geometrical references and humorous storyline, the book can be used with students of all ages.

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References:

------. 2000. Principles and Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics.

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